Soğurmanın çıkış sinyali üzerine etkisi

Abstract

İçsel sürtünmenin tüm etkİ6İ boyutsuz nitelik faktörü (Q) ve soğurma katsayısı O ile kayıp faktörü diye adlandırılan 1/Q ile ifade edilmişlerdir. ve Q arasındaki ilişki : Jl avT ile verilmiştir. Yeriçinde frekansla değişen ve lineer kabul edilen <*(/) eğrisi üzerinde yapılan tecrübeler aslında lineerlikten sap maların olduğunu göstermiştir. Belli frekans bantlarında daha az veya daha çok miktarda soğurma özelliğine sahip yapıların doğrusallıktan sapmanın nedeni olabileceği kabul edilmiştir. Şimdiye kadar taraftar bulan genel kabul ise bu tip oynamaların ölçüm hataları ve aletsel hatalardan kaynaklanabileceği üzeriney di. Bu amaçla faz spektrurou lineer olmayan bir flicker sinyalinin değişik frekans bantlarında aynı oranda soğurulmasmın veya aynı frekans bandında farklı oranlarda soğurulmanın çıkış sinyali üz erine etkileri araştırılmıştır. Doğrusal olarak artan soğurmanın yanında çeşitli nedenlerle belli frekans bantlarında aynı derecede, daha az veya daha çok soğurulmuş kısım, çıkış sinyaline farklı şekillerde etkimektedir. Farklı frekans bantlarında aynı oranda soğurma, sinyalin, genlik spektrumlarında artma ve azalmaya, çeşitli boyutlarda salınım- lara neden olduğu gibi lineer soğurma hali için elde edilen çıkış sinyaline göre daha geç veya erken de gelebilmektedir.

The major features of seismic wave propagation which have been observed experimentally would be expected on the basis of a purely elastic earth. The pattern of reflected and refracted body waves and dispersion of surface waves can all be derived by application of the equations of elasticity to media whose bound aries are chosen to conform to the section of earth involved. However, there are differences between observation and theoreti cal expectation, the principal one being a loss of amplitude in excess of that due to geometrical spreading and reflection at boundaries. This extra loss of amplitude will be called ATTENUA TION. The elastic properties of rock are uniquely defined by elas tic moduli and/or P and S wave velocities. Generally accepted definitions and units for these two parameters make their use commonplace. The attenuative properties of rocks, however, are specified by a wide range of measures. In order to compare at tenuation data properly from different sources, it is important to present definitions of the different measures and to show how they are related to one another. The most commonly used measures of attenuation found in the literature are the attenuation coefficient a which is the ex ponential decay constant of the amplitude of a plane wave travel ling in a homogeneous medium; the quality factor Q and its in verse Q"1.sometimes called the internal friction or dissipation factor; and the logarithmic decrement 6. These quantities are related as follows : VI 1 ^üm £ Q* nf" it where v is the velocity and f is the frequency. Since both velocity and attenuation are associated with a particular mode of wave propagation, one experimental technique may yield an exten- sional wave velocity controlled by the Young's modulus and a dissipation factor denoted Q~l, while another may determine the P wave velocity and. In general, these results are not equi valent. For plane waves, propagation in a homogneous medium, the amplitude is given by where w is the angular frequency and k is the wave number. Attenuation may be introduced mathematically by allowing either the frequency or wave number to be complex. In the latter case, so that A(x,t)-A0e-a*et{k*M) where a is the attenuation coefficient in units of inverse length and the phase velocity is w Attenuation may also be defined in terms of inverse time by VII allowing w to be complex. Letting the attenuation be determined by For an oscillating system in free decay, the definition of the logarithmic decrement follows gives : _. Mil, olv s-]n[TA'aK'T The most common measures of attenuation are the dimension- lees quality factor Q and its inverse Q~l, As an intrinsic prop erty of rock, Q is a ratio of stored energy to dissipated energy. O'Connell and Budiansky (1978) discussed in detail various defi nitions of Q and their relationships to the viscoelastic consti tutive equations for a given material. Intrinsic Q may differ under some conditions from Q values derived from processes such as wave propagation. Yet these pro cesses are valuable tools for measuring the anelastic response of a rock. The various definitions of Q are equivalent to intrinsic Q if losses are assumed to be small (Q>10). Fortunately, under most conditions of interest in geophysics, t?e smaJ dissi ation assumption is valid. Intrinsic Q may be defined as : wE ^2rtV Q~-dE/dr Ati VIII where E is the instantaneous enei-gy in the system, dE/dt is the rate of energy loss, W is the elastic energy stored at maximum stress and strain, and \AW is the energy loss (per cycle) of a harmonic excitation. For nearly elastic or low- loss linear solids, an alternative definition may be found from the stress-strain relations. Given a sinusoidally varying stress, the strain response will also be sinusoidal. The two are related by the appropriate elastic modu lus M and the phase lag $ of strain behind stress. Allowing M to be complex where it can be shown that : (White, 1965) _ " -- « tan A" 0 Q M* We talked about the several theoretical models which have been proposed to clarify the mechanism of energy loss. One widely investigated method of introducing losses has the advantage that it yields a linear wave equation which can be solved for arbitrary time dependence. The assumption is made that stresses are directly proportional to strain rates, as well as to the components of strain themselves. This assumption was introduced independently by Stokes, Kelvin and Voigt, and its implications have been investigated by many. This kind of medium will be called a Voigt solid. (White, 1983) The form of the disturbance actualy observed agreed very well with the form calculated for particular type of earth ab sorption. The particular type of absorbtion which the earth had IX to have in ordei* to account for the seiemograms obtained in practice, was such that the coefficient of absorbtion for contin uous sinusoidal waves was to be proportional to the square of the frequency. Also there was to be no dispersion. This kind of medium (viscoelastic medium) can be solved by Stokes equation. (Ricker, 1977) The frequency domain represantation of the input-output wavelets in the form we are presently using is : G(/)e*,"-//(/)eiM"/?elr/Ii(/)ea (/» H(f) and G(f) are the amplitude spectra, h(f) and g(f) are the input and output wavelets, respectively. R is a frequency- inde pendent real scale factor representing geometrical spreading, transmission and reflection coefficients and free surface ef fects; i.e., any constant that does not distort the wavelet shape. r is the frequency- independent phase-shift due to complex reflection coefficients which occur beyond critical incident an gles, if any. The causal-absorption coefficient is BV)-H[\nAV)] In this study, I investigated the departure from the assumed rectilinear behaviour of the frequency dependent absorbtion coef ficient, <*. The perturbation of the observed data around linear trend of the at is, in general, accounted for the uncertainties in the observations, the noise present in the data, or the local irregularities in the surrounding rocks. I propose, in this the- sis, the variations in the linear behaviour of a may have some physical bases rather than statistical scattering a the data. In micro level, the interractions among neigboring molecules con stituting the rocks cause the "solid-solid" friction which re sults in the transformation of the mechanical energy to other forms of energy (i.e., heat, dislocation, chemical, etc). The so lid- so lid friction accounts for the rectilinear behaviour of the e< In the macro level, the rock system stimulant force inter action is dominant. The relation between the naturel resonant frequency of the rock system and the dominant frequency of the simulant forces becomes important. I bel ive the latter may be the reason of the mentioned perturbation from the rectilinear shape of a. This, in turn, may be an important parameter in earthquake engineering. It is well known that at resonant fre quencies, the energy transfer from mechanical to heat, deforma tion, and rupture increases. Therefore, the investigation of such physical behaviours at small scale (with low energy artifi cial sources) may be answer to the prediction of the possible devistatihg effects of the earthquakes. This study shows in some detail, the time and frequency behaviours of such absorptive mechanisms based on mathematical models»