CONSTANT Q-WAVE PROPAGATION AND ATTENUATION
CONSTANT Q-WAVE PROPAGATION AND ATTENUATION
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DOI：
10.1029/jb084ib09p04737
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发表时间：
1979-01-01
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通讯作者：
KJARTANSSON, E
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文献类型：
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作者：
KJARTANSSON, E
提出了波衰减的线性模型，其中q，即在每个周期或波长期间损失的能量部分，与频率完全无关。波的传播完全由两个参数规定，例如，Qandc0，在任意参考频率ω0处的相速度。一个简单的精确推导可以得到相速度作为频率函数的表达式：c/c0= (ω/ω0)γ，其中γ = (1/π) tan−1（1/Q）。导出了脉冲传播的尺度关系，并表明对于给定q值的材料，脉冲的上升时间或宽度与传播时间完全成正比。由δ函数源atx= 0产生的脉冲的行程时间与toxβ成正比，其中β = 1/（1‐γ）。根据这一关系，我们认为与非弹性有关的速度色散在时域上比在频域上更容易观察到。Strick提出的最陡下降近似给出了脉冲响应的时域表示。该比例关系应用于科罗拉多Pierre页岩地层的现场观测，由Ricker和McDonal等人发表，他们用频率成反比的Voigt固体来解释他的数据，而McDonal等人则用非线性摩擦来解释他们的数据。常数理论适用于这两组数据。
A linear model for attenuation of waves is presented, withQ, or the portion of energy lost during each cycle or wavelength, exactly independent of frequency. The wave propagation is completely specified by two parameters, e.g.,Qandc0, a phase velocity at an arbitrary reference frequency ω0. A simple exact derivation leads to an expression for the phase velocitycas a function of frequency:c/c0= (ω/ω0)γ, where γ = (1/π) tan−1(1/Q). Scaling relationships for pulse propagation are derived and it is shown that for a material with a given value ofQ, the risetime or the width of the pulse is exactly proportional to travel time. The travel time for a pulse resulting from a delta function source atx= 0 is proportional toxβ, where β = 1/(1 ‐ γ). On the basis of this relation it is suggested that the velocity dispersion associated with anelasticity may be less ambiguously observed in the time domain than in the frequency domain. A steepest descent approximation derived by Strick gives a good time domain representation for the impulse response. The scaling relations are applied to field observations from the Pierre shale formation in Colorado, published by Ricker, who interpreted his data in terms of a Voigt solid withQinversely proportional to frequency, and McDonal et al., who interpreted their data in terms of nonlinear friction. The constantQtheory fits both sets of data.