Prediction of Earthquakes; Considering the Ratio of P-to-S Wave Velocities

The ratio of P-to-S-wave velocities (Vp/Vs) is considered as one of the important measures of stressed natural rocks. The changes of this ratio were observed in various earthquakes, however the time of occurrence of the earthquakes were not determined, since the predictions were not combined with other prediction methods. This discriminant provides information about formation and evolution of composition for the earth’s crust. In addition, the changes in Vp/Vs before and after earthquakes are the possibly reliable direction to realizing the mechanics and probably predicting earthquakes. However, due to lattice orientations of anisotropic minerals, most of earth’s rocks are anisotropic and there exist cracks and thin layerings. The changing patterns in Vp/Vs acquired from seismic propagations can be readily analyzed according to the ray path and the polarization of the (P) pressure and (S) shear waves. The variations in their ratios carry information about stress variations of THE crust of the earth. The arising stresses in rocks are the messengers of possible earthquakes. The stations with monitoring devices and Geographical Positioning System (GPS) upgraded WI-FI remote data acquisition system can serve monitoring stress and movements in the earth’s crust. In view present technological developments, the system is reasonable cost effective and prediction of earthquakes can be possible in narrowed time interval. The laboratory tests will be also part of the prediction process.


Introduction
Earthquakes are causing many devastations and deaths every year. Unfortunately, there is no scientific prediction methods does exist so far, Lomnitz [1], Hough [2], Gullu [3], Aggarwal et al. [4], Verdugo [5], Liu et al. [6], Sara [7] and Crampin et al. [8]. Earthquakes are sudden and unexpected events and predictions are far away from reality. Prediction of earthquakes and their determination of the epicenter are important to minimize their damage to dwellings on the earth. Few methods can be mentioned, such as watching historical earthquake data, microtremor data, emission of radon gas in the area, observing movement of animals, etc. In this paper, the change of ratio V p /V s , Cekirge [9], that is velocities of pressure and shear waves is explained while they are traveling through the earth's crust. These waves are carrying information from the earth's crust. These velocities are function of material constants of the medium in that the waves are travelling. If the stress arises and accumulates in the medium where these waves propagate, the material constants are being exposed to change. The detection of stress field, that causes the earthquake, brings very important information about occurrence of earthquakes, Passelègue et al. [10], Ohnaka [11] and Vassiliou and Kanamori [12]. As stated in Wang et al. [13]; the conventional criteria for V p /V s ratio are V p /V s ≤ 1.756 for felsic rocks, 1.756 < V p /V s ≤ 1.809 for intermediate rocks, 1.809 < V p /V s ≤ 1.944 for mafic rocks, and V p /V s > 1.944 fluidfilled porous/fractured or partially molten rocks. Seismic waves and their dynamic were studied in depth and the arising stress is the feature of these activities, Aki and Richards [14] and Lowrie [15]. There are abundant studies between seismic waves and earthquakes, Okamoto et al. [16] and Takemura et al. [17]. The seismic waves propagate in nonhomogeneous and anisotropic media; it is response of the stressed of the medium. Propagation patterns between P and S waves are the diagnostics of the medium where these waves propagate. These patterns are distorted with the rising stress field in the medium. The observation of these could be used as a stethoscope for the earth's crust.
The governing equations for the media where P and S waves propagate are presented Koh and Eringen [18], Green and Zerna [19] and Eringen [20]. The P and S waves are small amplitude waves propagating initially stressed media, the propagation of these small amplitude harmonic waves in an initially stressed media were developed by Hayes and Rivlin [21] for elastic; Flavin and Green [22] for viscoelastic; and Hayes and Rivlin [23], Cekirge and Suhubi [24], Suhubi [25] and Chadwick [26] for thermoelastic materials. The waves are propagating in a medium that is subjected to respectively large static deformations, the State D, Figure 1; with the seismic disturbances, the State S. These stresses are the cause of the changing material constants of the medium. It is assumed that the time dependent harmonic deformations superposed on the deformed state D. The disturbances are sufficiently small and second order terms are neglected while the governing equations are being developed. The propagation of harmonic plane waves is considered in this initially stressed compressible medium. Through perturbation techniques, the governing equations are being solved up to second order that is providing sufficient accuracy for the these seismic waves. The detailed analyses of the governing equations are given in these references.

Wave Propagation in Deformed State
In State D, the body in static equilibrium, where, T ij = stress components in State D, and i,j and k = indices. A further time dependent deformation is considered and the particle x k moves to x k * as, where, ε = perturbation variable, and t = time. The resulting state is State S, where stress field becomes T + ε t*, t* is the superposed stress field due small amplitude disturbances in State D, Green and Zerna [12] and Suhubi [8]. If we consider earth's crust isotropic and homogenous medium, the Hook's law can be written as, where, t ij * = stress components in State S, i, j and k = indices, e ij = deformation components, δ ij = Kronecker delta, (δ ij = 1, if i = j, and δ ij = 0 if i ≠ j), η and µ = material constants, and u i = the displacements in State S along x i directions, i = 1, 2 and 3.
It should be mentioned that the material constants are being changed due stress filed T. The deformations in term of displacements u i 's, which are particle movements in State S, where, e ij = strain components. With initial strains, Equation (1) is written as, ρ (∂ 2 u i /∂t 2 ) = ∂t ij * /∂x i + u i,jk T jk + u j,jk T ik (5) where, u i = displacements from undeformed state to State S, ρ = material density in State D, and t = time.
where, v s = wave velocity of transvers waves, S waves. c) Surface waves; where, v sur = wave velocity, which is a plane wave propagating with the speed of v sur in x 3 direction.
where, A 1 and B 1 = amplitude of Rayleigh waves, b = attenuation constant for the Rayleigh waves, v r = velocity of Rayleigh waves, and i 2 = -1.
The relation between v r , v p an v s is written as, and the relation between the ratios of v r /v p , v s /v p and v r /v s to Poisson ratio, which is material constant, is presented by Figure 4. e) Love Surface waves where, A 3 = Amplitude of Love waves, b = Attenuation constant for the Love waves, and v L = velocity of Love waves.
where, µ = material constants in semi-infinite medium, µ1 = material constant in the layer, vs = velocity of s wave in semi-infinite medium, vs 1 = velocity s waves in the layer, and H 1 = thickness of the layer.
Since displacements are too small, and as a result, the material density is considered as constant. Then, equation (2.20) is separated for presenting the following waves, and In these equations, and v s = (µ/ρ) 1/2 .
These wave velocities are function of material constant in State D, where the material constants are differing respect to material constants in undeformed state. The comparative studies can generate information for arising stress field and ultimately, possible earthquakes. Of course, the predictions require comparative studies for arising stress field.

Rate of Changing Stress Field
The idea is the not only for determining the arising stress field, also evolution of the rate of arising field since the stress field is being monitored and recorded in the field as time series. The failure criteria can determine the breaking point of the material. Since the rate of stress field is monitored, the time of the failure or the earthquake could be predicted or determined at time within an interval of tolerance.
Since time dependent behavior for stress field is monitored, it is possible determining the time of breaking or failure time of the formation. The material may reach the failure point at predicted or determined time; that is the earthquake. The material failure theories for ductile and brittle materials such as maximum shearing stress, (Tresca), maximum distortion energy (Von Mises), maximum normal stress (Coulomb) and maximum normal strain (Saint-Venant) theories or combination of these could be used, Beer et al. [28], Christensen [29], Timoshenko [30], Hearn [31], Makhlouf and AliofKhazraei [32] and Buehler [33]. The combination of these theories could be used to determine failure time by considering material structure of the area under consideration. The randomness of material constants in the geological structures will also be accounted for determining expected time of earthquake with its variance. This consideration will result a time interval of occurrence of the earthquake with a confidence. The key factor is determining arising stress field and its growing rate. These criteria, of course, should be supported by field observations, material samplings and laboratory studies. These methodologies are aiming at the predicting time and shortening time interval of occurrences of earthquakes. It should be noted that there is a tolerance cushion of time in these predictions.

Conclusions
Seismic disturbances are being observed by the stations that can estimate velocities of P and S waves. These values and their ratios are containing in information from depths of the earth's crust. Changes in these values are the messengers of the seismic activities in the earth's crust. These values can be compared with sample values taken from the earth's crust, the compared values are material constants and wave velocities. It is necessary to establish a network to observe these waves, the movements earth's can be measured in these stations as well. Of course, several methods should be used for the prediction of earthquakes accompanied with geology of the area. The heterogeneity, anisotropy and layered structure exist in the medium, and these factors less effective than information of the stress that is carried by the infinite small amplitude waves.
The information about structure of the area is respectively unchanging situation in the medium. Essentially, these waves carry information about the changing and arising stress field in the medium. A network of stress monitoring with Geographical Positioning System (GPS) stations around earthquakes zones could be cost effective since this system can utilize a remote WI-FI data acquisition system. In these stations, the stress field and earth movements can be monitored around the clock for predicting earthquakes.