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• W4224262472 abstract "The power-law dependence of the quality factor <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Q</mml:mi> </mml:math> on frequency (i.e. <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:msup> <mml:mrow> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> </mml:msup> </mml:math> ), which is called the power-law frequency-dependent <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Q</mml:mi> </mml:math> for brevity, is a common phenomenological description of seismic wave attenuation in the interior of the Earth. The wave equation in differential form is essential to forward and inverse modelling of dissipative seismic waveforms in an accurate and efficient manner. However, all existing methods are seemingly unable to explicitly incorporate the exponent parameter <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> into the wave equation in differential form. This drawback apparently limits the development of gradient-based inverse methods for spatially varying <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> via the wave equation. In this paper, we use a weighting function method to derive the viscoacoustic wave equations for the power-law frequency-dependent <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Q</mml:mi> </mml:math> , which explicitly involve the parameters <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> . A critical step in this method is to construct a dissipative model, for which the complex modulus is expressed as an <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>N</mml:mi> </mml:math> -order series in terms of the weighting function. Numerical examples are used to illustrate the accuracy of the dissipative model, the effect of <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>γ</mml:mi> </mml:mrow> </mml:math> on waveforms, and the application of the newly proposed wave equations in viscoacoustic wavefield modelling for the power-law frequency-dependent <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Q</mml:mi> </mml:math> ." @default.
• W4224262472 created "2022-04-26" @default.
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• W4224262472 date "2022-04-01" @default.
• W4224262472 modified "2023-09-26" @default.
• W4224262472 title "Viscoacoustic wave equations for the power-law dependence of <i>Q</i> on frequency" @default.
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• W4224262472 doi "https://doi.org/10.1098/rspa.2022.0024" @default.
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